Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition $$ \widehat{A} := \underset{n}\varprojlim A/I^n. $$ This operation is well known and has a lot of good properties, for example it is $I$-adically complete.

In an article I'm reading I found the sentence "completion at all prime ideals such that...". My question is: what is the completion with respect to a family of ideals?

My guess is the following: suppose we have finetely many ideals, say $I_1,\ldots,I_n$. Then the final result is obtained taking the completion wrt $I_1$, then the completion wrt (the ideal generated by) $I_2$ and so on. If we have infinitely many ideals we have to take a direct limit.

Is this construction discussed somewhere? Has it good properties? For example it's not totally clear to me that the order of the ideals doesn't affect the final result.

Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition $$ \widehat{A} := \underset{\leftarrow}\lim A/I^n. $$ This operation is well known and has a lot of good properties, for example it is $I$-adically complete.

In an article I'm reading I found the sentence "completion at all prime ideals such that...". My question is: what is the completion with respect to a family of ideals?

My guess is the following: suppose we have finetely many ideals, say $I_1,\ldots,I_n$. Then the final result is obtained taking the completion wrt $I_1$, then the completion wrt (the ideal generated by) $I_2$ and so on. If we have infinitely many ideals we have to take a direct limit.

Is this construction discussed somewhere? Has it good properties? For example it's not totally clear to me that the order of the ideals doesn't affect the final result.